Algorithmic thresholds for tensor PCA

Gerard Ben Arous, Reza Gheissari, Aukosh Jagannath

Abstract:

We study the algorithmic thresholds for principal component analysis of
Gaussian k-tensors with a planted rank-one spike, via Langevin dynamics and
gradient descent. In order to efficiently recover the spike from natural initial-
izations, the signal-to-noise ratio must diverge in the dimension. Our proof
shows that the mechanism for the success/failure of recovery is the strength
of the “curvature” of the spike on the maximum entropy region of the ini-
tial data. To demonstrate this, we study the dynamics on a generalized family
of high-dimensional landscapes with planted signals, containing the spiked
tensor models as specific instances. We identify thresholds of signal-to-noise
ratios above which order 1 time recovery succeeds; in the case of the spiked
tensor model, these match the thresholds conjectured for algorithms such as
approximate message passing. Below these thresholds, where the curvature of
the signal on the maximal entropy region is weak, we show that recovery from
certain natural initializations takes at least stretched exponential time. Our ap-
proach combines global regularity estimates for spin glasses with pointwise
estimates to study the recovery problem by a perturbative approach.